Linked Questions

10
votes
5answers
4k views

Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?

All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before. Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
13
votes
5answers
341 views

Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
11
votes
3answers
5k views

Energy momentum tensor from Noether's theorem

in the book Quantum Field Theory by Itzykson and Zuber the following derivation for the stress-energy tensor is proposed (p.22): Assume a Lagrangian density depending on the spacetime coordinates $x$ ...
14
votes
1answer
4k views

Trick for deriving the stress tensor in any theory

In D. Tong's notes on string theory (pdf) section 4.1.1 he explains a trick for deriving the stress-energy tensor which arises from translations in the base manifold of the field theory (in this case ...
9
votes
1answer
1k views

Why is the Hamiltonian zero in relativity?

I'm trying to understand something with the lagrangian and the hamiltonian formalisms in relativity theory, and why the following result cannot be the same in classical (non-relativistic) mechanics. ...
3
votes
2answers
1k views

Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
4
votes
1answer
444 views

How to calculate an axial anomaly in 1+1 dimensions?

As far as I understand, an axial $U(1)$ transformation transforms a two-component spinor like $$ \psi \to \psi'=\text e^{\text i\epsilon \gamma^5 }\psi,\qquad \psi=\begin{pmatrix}\psi_1\\\psi_2\end{...
6
votes
3answers
526 views

Derivation of the Noether current

(Cf. Di Francesco et al, Conformal Field Theory, pp. 40-41) I am trying to derive eqn. (2.142) or $$\delta S = \int d^d x ~\omega_a~\partial_{\mu}j^{\mu}_a \tag{2.142}$$ in the book CFT by Di ...
2
votes
1answer
309 views

Subtlety in derivation of Noether's theorem by Di Francesco

In the book 'Conformal Field Theory' by Di Francesco et al, a derivation of Noether's theorem is demonstrated by imposing that, what I believe is said to be a more elegant approach, the parameter $\...
2
votes
2answers
201 views

Noether's Theorem in Classical Field theory Confusion

Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\mathcal{L} = \mathcal{L}(φ_i, \partial_μφ_i).$$ Suppose we perform the following infinitesimal ...
1
vote
1answer
271 views

Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem

Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates : $$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}...
3
votes
1answer
188 views

Conserved currents in Noether Theorem with varying parameter

I have a continuous transformation on the field $\phi$ of the form $$\phi(x)\rightarrow \phi'(x)=\phi(x)+\alpha\Delta\phi(x),\tag{1}$$ where $\alpha$ is a constant infinitesimal parameter and $\...
5
votes
1answer
86 views

What is the meaning of the parameter in Noether's theorem?

According to the explanation of Noether's theorem in Peskin & Schroeder's QFT book, pp. 17-18, If the Lagrangian $\mathcal{L}(x)$ change to $$\mathcal{L}(x)+\alpha\partial_\mu\mathcal{J}^\mu\tag{...
3
votes
0answers
268 views

Global and local symmetries in Noether's theorem. And also Stress-Energy tensors

Noether's theorem for fields is usually given as follows: Given a field theory with action $S=\int\mathcal{L}(\phi,\partial\phi)d^4x$, and given a one-parameter variation of the fields $\phi_\epsilon$...
1
vote
2answers
86 views

Noether's theorem under arbitrary coordinate transformation

Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. Suppose our action is of the form $S = \int d^4x\, \mathcal{L}(\...

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